Convergence to Gibbs equilibrium - unveiling the mystery

We consider general hamiltonian systems with quadratic interaction potential and $N<\infty$ degrees of freedom, only $m$ of which have contact with external world, that is subjected to damping and random stationary external forces. We show that, as $t\to\infty$, already for $m=1$, the unique limiting distribution exists for almost all interactions. Moreover, it is Gibbs if the external force is the white noise, but typically not Gibbs for gaussian processes with smooth trajectories. This conclusion survives also in the thermodynamic limit $N\to\infty$.